Asset Pricing with Nonlinear Principal Components

Working paper, 2021

Motivation

Look for a parsimonious stochastic discount factor (SDF);
Increasing number of factors explaining the cross-section (CS) (Factor zoo.)
Kozak et al. (2020) show the importance of rotating the SDF into a transformed space.
Prior literature: Rotate the SDF into the space of linear principal components (PCs);
This paper: Rotate the SDF into the space of nonlinear principal components;

First paper to empirically test the effectiveness truly independent nonlinear factors
- In an asset pricing involving the identification of an SDF that prices the CS of stocks.

Let $ r_{t}=( r_{1,t},…, r_{N,t})’$ be the vector of excess returns of N portfolios, t=1,…,T
Let $Z_{t}$ be a N-by-k matrix of asset anomaly characteristics;
Let $RC_t=Z_{t-1}’r_t$ be a k-by-1 vector of raw characteristic returns;
- Extract linear and nonlinear factors from RC;
- F is either RC or the linear factors or the nonlinear factors.
Let $\Sigma=Cov(F)$ be a k-by-k variance-covariance matrix of the factors;
Let $\mu=\mathbb{E}(F)$ be a k-by-1 vector of expected factor returns;
$SDF_t=1-\lambda’(F_t-\mathbb{E}F_t)$
We estimate two models : a purely linear model and an hybrid model;
For each model :
We impose two kind of penalties to estimate the SDF coefficients :
- L2 pen : $\hat \lambda=arg \min_{\lambda}( \mu -\Sigma \lambda)’\Sigma^{-1}( \mu -\Sigma \lambda)+\gamma \lambda’\lambda$
- L1L2pen : $\hat \lambda=arg \min_{\lambda}( \mu -\Sigma \lambda)’\Sigma^{-1}( \mu -\Sigma \lambda)+\gamma_1 \sum_{i=1}^{k} |\lambda_i |+\gamma_2\lambda’\lambda$
Estimate the parameter $\hat \lambda$ via Ridge or Elastic net using LAR-EN;
Choose optimally the tuning parameters $\gamma$ or ( $\gamma_1$ and $\gamma_2$) using out-of-sample $R^2$.

For different fixed cross-sections of returns, the nonlinear SDF consistently outperforms the linear specification;
For the FF25P: 65% versus 49%
For the 50 anomalies: 55% versus 22%
Nonlinear SDF requires less factors.
For the 50 anomalies: 5 factors versus 15-20 factors